A proposition from Fulton's book 'Algebraic Curves':
Let $I$ be an ideal in $k[X_1, …, X_n]$ ($k$ algebraically closed), and
suppose $V(I) = \{ P_1, \ldots, P_N \}$ is finite. Let
$\mathcal{O}_i = \mathcal{O}_{P_i}(\mathbb{A}^n)$. Then there is a natural
isomorphism of $k[X_1, \ldots, X_n] \mathbin{/} I$ with
$\prod_{i = 1}^{N} \mathcal{O}_i \mathbin{/} I\mathcal{O}_i$.
But if $k$ is not algebraically closed, is it still true that, if $V(I)$ is
finite, $k[X_1, \ldots, X_n] \mathbin{/} I$ is a direct product of local
rings? And if so, then some factors are most likely given by localizations at
the points of $V(I)$, whereas other factors are not; how to describe these
latter factors?
The reason for this question is the observation that field extensions of $k$
are fields and thus local rings. For instance,
$\mathbb{Q}[X] \mathbin{/} (X^2 + 1)$ is a field, but $V(I)$ is empty and
$\mathbb{Q}$ is not algebraically closed. Hence, by the Chinese remainder
theorem, it seems to me that, for any $f$ in $k[X]$, $k[X] \mathbin{/} (f)$ is
the direct product of local rings anyway. Is it at least that correct?
Best Answer
If you upgrade to schemes and take $V(I)$ to be the closed subscheme $\operatorname{Spec} k[X_1,\cdots,X_n]/I \subset \operatorname{Spec} k[X_1,\cdots,X_n] = \Bbb A^n_k$, then your claim that $V(I)$ finite implies a natural isomorphism between $k[X_1,\cdots,X_n]/I$ and $\prod \mathcal{O}_i/I\mathcal{O}_i$ is still true. Without upgrading to schemes or enforcing the requirement that $k$ is algebraically closed, you cannot guarantee this result will still hold.
In the case of a scheme, the reason is that the former is the global sections of the structure sheaf on $V(I)$, while the latter is the product of the stalks, and on any discrete space these two things are naturally isomorphic. The sheaf condition gives that the global sections are the product of sections over each open set $\{p_i\}$, while the fact that the $\{p_i\}$ are open means they're the final object among open sets containing $p_i$ and thus the stalk at $p_i$ is the value of the sheaf on $\{p_i\}$.